A New Approach to Incremental Cycle Detection and Related Problems

We consider the problem of detecting a cycle in a directed graph that grows by arc insertions, and the related problems of maintaining a topological order and the strong components of such a graph. For these problems we give two algorithms, one suited to sparse graphs, the other to dense graphs. The former takes O(min{m1/2, n2/3}m) time to insert m arcs into an n-vertex graph; the latter takes O(n2 log n) time. Our sparse algorithm is substantially simpler than a previous O(m3/2)-time algorithm; it is also faster on graphs of sufficient density. The time bound of our dense algorithm beats the previously best time bound of O(n5/2) for dense graphs. Our algorithms rely for their efficiency on vertex numberings weakly consistent with topological order: we allow ties. Bounds on the size of the numbers give bounds on running time. Key words. topological ordering, cycle detection, strongly connected components, incremental data structure AMS subject classifications. 68P05, 68Q25, 68R10, 68W...

• Type: Journal
• Year: 2016
• Where: TALG
• Authors: Michael A. Bender, Jeremy T. Fineman, Seth Gilbert, Robert E. Tarjan
• Keywords: TALG 2016